In the multi-linear regression, how is the value of weight $b_2$ calculated?
It is calculated the same way $b_1$ is calculated. 

Nearly following your notation, say your multiple linear regression function is 

$H(X_i) =  b_0 + b_1x_{1,i} + ...+ b_nx_{n, i}$

for data instance $X_i=x_{1,i},...,x_{n, i}$
and weights $b_0,...,b_n$.

And say your error function is $E(X,Y) = \sum_i(H(X_i)-Y_i)^2$

where $X$ is the collection of all data points $X_i, Y_i$.

From your error function $E$, for whatever weights you have (with a gradient based method), calculate the partial derivative
$\partial E /\partial b_i$ and use this to update all of your weights at once in each iteration of your optimization routine.